Path-Dependent Currency Options With Mean Reversion Hoi Ying Wong∗, Ka Yung Lau Department of Statistics The Chinese University of Hong Kong, Hong Kong. March 18, 2007
∗
Corresponding author; fax: (852) 2603-5188; e-mail:
[email protected].
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Electronic Electronic copy of copy this paper available is available at: http://ssrn.com/abstract=975012 at: http://ssrn.com/abstract=975012
Path-Dependent Currency Options With Mean Reversion Abstract This paper develops a path-dependent currency option pricing framework in which the exchange rate follows a mean-reverting lognormal process. Analytical solutions are derived for barrier options with a constant barrier, lookback options and turbo warrants. As the analytical solutions are obtained in Laplace transform, we show numerically that our solution implemented with a numerical Laplace inversion is efficient and accurate. The pricing behavior of path-dependent options with mean-reversion is contrasted to the Black-Scholes model.
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Electronic Electronic copy of copy this paper available is available at: http://ssrn.com/abstract=975012 at: http://ssrn.com/abstract=975012
1 Introduction The currency option market can rightfully claim to be the world’s only truly global, 24-hour option market. The currency option market is among the largest of the option markets by trading volume. The most recent BIS survey estimated that the outstanding notional amount of OTC currency options was $9,027 billion in face value in June 2006. Garman and Kohlhagen (1983) proposes a convenient approach of using the Black-Scholes formulas to value call and put currency options but the BlackScholes model is far away from satisfactory for pricing path-dependent options because currencies are well known to have mean reverting effect. Ekvall et al. (1997) give several reasons for mean-reverting exchange rates using an equilibrium model. Sweeny (2006) gives empirical evidence for mean reversion in G-10 nominal exchange rates. For more evidence, we refer to Hui and Lo (2006) and the references within their paper. Sorensen (1997) proposes a mean-reverting process for currency through the dynamics in the domestic and foreign term structure of interest rates. Although the mean-reverting feature has a little effect on vanilla currency options, Sorensen (1997) finds that it significantly affects American currency option prices. As American options can be regarded as path-dependent options, it is believed that the mean-reverting feature affects other path-dependent options as well. The foreign exchange market is a fertile ground for the invention of new exotic options. The most popular types of exotic options are path dependent options such as barrier options, lookback options and turbo warrants. Hui and Lo (2006) employ a mean-reverting lognormal (MRL) model to study the pricing behavior of options with a soft barrier, which is a deterministic function of time that is specifically constructed to match with the coefficients of the governing partial different equation of the barrier option. Hui and Lo (2006) find that the parameters in the MRL model have a material impact on the valuation and hedging parameters of barrier options. One type of barrier option is the knock-out option. A knock-out option is similar to a vanilla option except for the existence of a barrier exchange rate, called the out-strike, which when breached would cause the option to extinguish at any time during the option’s life. In practice, the barrier is usually (if not always) chosen to be a fixed constant, instead of a soft barrier. To facilitate this practical need, Hui and Lo (2006) simulate the fixed barrier as a slowly fluctuating soft barrier with small oscillating amplitude by tuning a parameter in the model. Approximated option prices are then obtained. To supplement their results, the first objective 3
of this paper is to derive an analytical barrier option pricing formula with a constant barrier under the MRL model. Although the solution is expressed in Laplace transform, the computation is very efficient as it takes less than 1 second to obtain an accurate option price. Our simulation shows that ours is more accurate than the approximation of Hui and Lo (2006) for constant barrier options. Another important feature of knock-out options is the rebate paid to option holders. The rebate is usually a predetermined amount of cash paid at the first time that the barrier is hit. Thus, the present value of this rebate is related to the moment generating function of the first passage time under the MRL process. We derive analytical solution for the present value of the rebate in barrier options. Lookback options are popular path-dependent options in the currency option market too. They provide opportunities for the holders to realize attractive gains in the event of substantial movements of exchange rates. For instance, the floating strike lookback call allows the holder to purchase the underlying exchange rate with the strike set as the minimum exchange rate over a given period. Investors who speculate on exchange rate volatility may be interested in the lookback spread option, the payoff of which depends on the difference between the maximum and minimum of exchange rates over a time horizon. More exotic forms of lookback payoffs are discussed by He et al. (1998). Wong and Chan (2007a) show that lookback option features are embedded in dynamic fund protection of insurance products. If the underlying fund follows the MRL process, then the result obtained in the present paper becomes useful. Our second objective is to derive analytical solutions for lookback options under the MRL model. Our third objective is to analytically determine the price for turbo warrants under mean reversion. Turbo warrants first appeared in Germany in late 2001 but they were nothing else than standard knock-out barrier options. A more interesting situation appears when the barrier is set to be strictly in the money and a rebate is paid if the barrier is hit. The rebate is usually calculated according to an exotic option payoff. At the end of February 2005, Societe Generale (SG) listed the first 40 turbo warrants on the Nordic Growth Market (NGM) and Nordic Derivatives Exchange. During February 2005, the turbo warrant trading revenue was 31 million kronor, 50% (31/55) of total NGM trading revenue of 55 million kronor. In June 2006, the Hong Kong Exchange and Clearing Limited (HKEx) introduced callable bull/bear contracts (CBBC), which are actually turbo warrants. There are two types of CBBC: N and R. The N-CBBC pays no rebate when the barrier is hit whereas the R-CBBC pays an exotic rebate, see HKEx (2006). The underlying asset of turbo warrants can either be equity or currency. Turbo warrants are attractive because it is believed that their prices are lower 4
than their vanilla counterparts and are much less sensitive to the implied volatility. Thus, investors can simply bet on upward or downward movement of an asset with a lower cost and minimal volatility risk. Eriksson (2005) derives explicit solutions to turbo call and put warrants using the Black-Scholes model. The rebate of the turbo put (call) warrant is the difference between the highest (lowest) recorded stock price during a pre-specified period after the barrier is hit and the strike price. Therefore, the rebate can be viewed as a non-standard lookback option. These turbo call and put options are essentially the R-CBBC in Hong Kong. Wong and Chan (2007b) study the turbo warrant pricing under several stochastic volatility models. Wong and Lau (2007) obtain an analytical solution for turbo warrants under a jump diffusion model. However, all these works concentrate on equity options and do not take into account the mean-reverting feature of exchange rates. The final objective of this paper is to investigate the impact of mean-reversion on the aforementioned path-dependent options, except for the barrier option as it has been studied by Hui and Lo (2006). We contrast the pricing behaviors between the Black-Scholes model and the MRL model. We find that the mean-reverting feature has significant effect to all exotic currency options considered in this paper. Although we concentrate on mean-reverting exchange rates, our results are also applicable to other asset classes with mean-reversion, such as commodity. The rest of the paper is organized as follows. Section 2 introduces the MRL model, derives moment generating functions for a mean-reverting process and obtains closed form solution for vanilla options. Barrier option pricing is studying in Section 3 where the present value of the rebate is obtained. Using simulation with many asset price paths as a benchmark, we compare the accuracy between the approach of Hui and Lo (2006) and ours. Section 4 derives analytical solutions to lookback options and examine the impact of mean-reversion on lookback option pricing numerically. Section 5 introduces turbo warrants and their pricing under mean-reversion. Section 6 concludes the paper.
2 The Model In the MRL model, it is assumed that the risk neutral dynamic of the exchange rate F (i.e., the domestic currency value of a unit of foreign currency) follows dF = [κ(ln F − ln F ) + (r − rf )]dt + σdW, F
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(1)
where F is the conditional mean exchange rate, κ is the speed of reversion, σ is the volatility of the exchange rate, r is the domestic interest rate, rf is the foreign interest rate, and W is the Wiener process. For the relation between the physical and risk neutral processes, we refer to Hui and Lo (2006) and the references within their paper. Let x = ln F and θ = [ln(F − σ 2 /2κ) + (r − rf )/κ]. Applying Itˆo’s lemma on x with respect to (1) yields, F = ex , dx = κ(θ − x)dt + σdW, x0 = ln F0 ,
(2)
where F0 is the initial exchange rate.
2.1 Moment Generating Functions Given the dynamic of the underlying asset, it is possible to obtain the moment generating function (MGF) for the log-asset value. Denote the MGF conditional on the information up to time t as M(ξ, x, T ), ie. M(ξ, x, T ) = E eξxT xt = x ,
where T ≥ t. Then, the following lemma holds. Lemma 2.1.
−κ(T −t)
M(ξ, x, T ) = exp ξxe
−κ(T −t)
+ ξθ 1 − e
ξ 2σ2 −2κ(T −t) . 1−e + 4κ
Proof. We solve (2) and obtain that σ 2 e−2κT 2κT −κT −κT xT ∼ N x0 e + (1 − e )θ, e −1 . 2κ Because xT is normally distributed, its MGF is available and is presented in the Lemma. As we are interested in path-dependent options that involve first passage time(s). We define the first passage times as τH+ = inf{t|Ft ≥ H} and τH− = inf{t|Ft ≤ H}.
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To simplify matters, we concentrate on the first passage time τH+ and omit the superscript ‘+’ throughout the paper. The MGF for the first passage time is presented in Lemma 2.2, where we use the notation MH (α) = E e−ατH x0 = ln F0 < ln H to denote the MGF. Lemma 2.2. Φ A, 21 , B12 + Φ A + 21 , 23 , B12 2Γ(A+1/2) B1 Γ(A) MH (α) = , B Φ A, 12 , B22 + Φ A + 21 , 23 , B22 2Γ(A+1/2) 2 Γ(A)
where Γ(·) is the gamma function, Φ(·, ·, ·) is the degenerated hypergeometric function, and √ √ α κ κ , B1 = [ln F0 − θ] , B2 = [ln H − θ] . A = 2κ σ σ Proof. See Appendix A, which also provides an expression for the MGF of τH− and introduces some basic properties of the degenerated hypergeometric function. The degenerated hypergeometric functions are also known as the confluent hypergeometric functions or Whittaker functions available in Mathematica and Matlab computer systems. These mathematical functions are widely applied in mathematics, physics and engineering. In the option pricing literature, these functions appear in, but not limited to, Davydov and Linetsky (2001), and Wong (2007) for the Laplace transform of first hitting times of the CEV model. We will see shortly that the MGFs, M(ξ, x, T ) and MH (α), are useful for path-dependent option pricing with mean reversion.
2.2 Vanilla Call Price Although the closed form solution for vanilla call option with mean reversion has been obtained by Ekvall et al. (1997), we would like to demonstrate the use of MGF for pricing vanilla call option via Laplace transform. The result obtained here will be useful later for barrier options. The vanilla call option has the payoff: C(T ) = max(FT − K, 0). Let k = − ln K and Lk,ζ denote the Laplace transform operator with respect to k. The call option price can be calculated by the following proposition. 7
Proposition 2.1. C(F, K, T ) =
L−1 k,ζ
e−rT M(1 + ζ, ln F, T ) , ζ(1 + ζ)
(3)
where the MGF, M(ξ, x, T ), is obtained in Lemma 2.1. Proof. Consider the Laplace transform: Z ∞ e−rT E[FTζ+1 ] e−rT −rT e e−ζk E(FT − e−k )+ dk = = E[e(ζ+1)xT ]. ζ(ζ + 1) ζ(ζ + 1) −∞ The last term is exactly the MGF defined in Lemma 2.1, with ξ = 1 + ζ. As the MGF, M(ξ, x, T ), is an exponential function with a quadratic exponent of ξ, the Laplace inversion in Proposition 2.1 can be explicitly obtained from the Laplace transform table. Specifically, it is easy to show that C(F, K, T ) = Fee−rT N(d1 ) − Ke−rT N(d2 ),
(4)
where N(·) is the standard normal cumulative distribution function (cdf) and σ 2 e−2κT 2κT −κT e −1 , Fe = F e exp(θ(1 − e−κT ) + ν 2 /2), ν 2 = 2κ ln(Fe/K) + ν 2 /2 d1 = , d2 = d1 − ν. ν The put price can then be implied by the put-call parity.
3 Barrier Option This section is devoted to barrier option pricing in which the barrier is a preselected constant. We focus on up-and-in (UI) options although other barrier options can be valued using the same framework. As a specific example, the up-and-in call (UIC) has the payoff: UIC(T ) = (FT − K)+ 1{τH
